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    "# 圆周运动的物理公式\n",
    "\n",
    "## 1. 基本运动学公式\n",
    "\n",
    "### 角位移\n",
    "$$ \\theta = \\omega t + \\frac{1}{2}\\alpha t^2 $$\n",
    "\n",
    "### 角速度\n",
    "$$ \\omega = \\frac{d\\theta}{dt} = \\frac{2\\pi}{T} = 2\\pi f $$\n",
    "\n",
    "### 角加速度\n",
    "$$ \\alpha = \\frac{d\\omega}{dt} = \\frac{d^2\\theta}{dt^2} $$\n",
    "\n",
    "### 线速度与角速度关系\n",
    "$$ v = \\omega r $$\n",
    "\n",
    "### 线加速度与角加速度关系\n",
    "$$ a_t = \\alpha r $$\n",
    "\n",
    "## 2. 匀速圆周运动\n",
    "\n",
    "### 向心加速度\n",
    "$$ a_c = \\frac{v^2}{r} = \\omega^2 r = \\frac{4\\pi^2 r}{T^2} $$\n",
    "\n",
    "### 向心力\n",
    "$$ F_c = m a_c = m\\frac{v^2}{r} = m\\omega^2 r $$\n",
    "\n",
    "### 周期\n",
    "$$ T = \\frac{2\\pi r}{v} = \\frac{2\\pi}{\\omega} $$\n",
    "\n",
    "### 频率\n",
    "$$ f = \\frac{1}{T} = \\frac{\\omega}{2\\pi} $$\n",
    "\n",
    "## 3. 变速圆周运动\n",
    "\n",
    "### 切向加速度\n",
    "$$ a_t = \\frac{dv}{dt} = \\alpha r $$\n",
    "\n",
    "### 法向加速度（向心加速度）\n",
    "$$ a_n = \\frac{v^2}{r} = \\omega^2 r $$\n",
    "\n",
    "### 总加速度\n",
    "$$ a = \\sqrt{a_t^2 + a_n^2} $$\n",
    "\n",
    "### 加速度方向\n",
    "$$ \\tan\\phi = \\frac{a_t}{a_n} $$\n",
    "\n",
    "## 4. 动力学公式\n",
    "\n",
    "### 牛顿第二定律在圆周运动中的应用\n",
    "$$ \\sum F_r = m\\frac{v^2}{r} = m\\omega^2 r $$\n",
    "\n",
    "### 常见向心力来源：\n",
    "- **重力**：$ F_g = G\\frac{Mm}{r^2} $\n",
    "- **弹力**：$ F_k = kx $\n",
    "- **摩擦力**：$ F_f = \\mu N $\n",
    "- **张力**：$ F_T $\n",
    "\n",
    "## 5. 能量关系\n",
    "\n",
    "### 动能\n",
    "$$ E_k = \\frac{1}{2}mv^2 = \\frac{1}{2}m\\omega^2 r^2 $$\n",
    "\n",
    "### 势能（重力场中）\n",
    "$$ E_p = mgh $$\n",
    "\n",
    "### 机械能守恒\n",
    "$$ E = E_k + E_p = \\text{常数} $$\n",
    "\n",
    "## 6. 特殊圆周运动\n",
    "\n",
    "### 圆锥摆\n",
    "$$ T = 2\\pi\\sqrt{\\frac{L\\cos\\theta}{g}} $$\n",
    "\n",
    "### 车辆转弯\n",
    "$$ v_{\\text{max}} = \\sqrt{\\mu g r} $$\n",
    "\n",
    "### 天体圆周运动\n",
    "$$ v = \\sqrt{\\frac{GM}{r}} $$\n",
    "$$ T = 2\\pi\\sqrt{\\frac{r^3}{GM}} $$\n",
    "\n",
    "## 7. 矢量表示\n",
    "\n",
    "### 位置矢量\n",
    "$$ \\vec{r} = r(\\cos\\theta\\hat{i} + \\sin\\theta\\hat{j}) $$\n",
    "\n",
    "### 速度矢量\n",
    "$$ \\vec{v} = \\frac{d\\vec{r}}{dt} = r\\omega(-\\sin\\theta\\hat{i} + \\cos\\theta\\hat{j}) $$\n",
    "\n",
    "### 加速度矢量\n",
    "$$ \\vec{a} = -r\\omega^2(\\cos\\theta\\hat{i} + \\sin\\theta\\hat{j}) + r\\alpha(-\\sin\\theta\\hat{i} + \\cos\\theta\\hat{j}) $$\n",
    "\n",
    "---\n",
    "\n",
    "## 📊 重要参数说明\n",
    "\n",
    "| 符号 | 物理量 | 单位 |\n",
    "|------|--------|------|\n",
    "| $ \\theta $ | 角位移 | rad |\n",
    "| $ \\omega $ | 角速度 | rad/s |\n",
    "| $ \\alpha $ | 角加速度 | rad/s² |\n",
    "| $ v $ | 线速度 | m/s |\n",
    "| $ a_c $ | 向心加速度 | m/s² |\n",
    "| $ F_c $ | 向心力 | N |\n",
    "| $ T $ | 周期 | s |\n",
    "| $ f $ | 频率 | Hz |\n",
    "| $ r $ | 半径 | m |\n",
    "\n",
    "## 💡 重要说明\n",
    "\n",
    "1. **匀速圆周运动**：速度大小不变，方向时刻改变\n",
    "2. **向心加速度**：始终指向圆心，改变速度方向\n",
    "3. **切向加速度**：改变速度大小\n",
    "4. **角量守恒**：在不受外力矩作用时，角动量守恒\n",
    "\n",
    "> 这些公式是分析圆周运动的基础，广泛应用于力学、天文学、工程学等领域。"
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